A Frequency Domain Analysis of the Excitability and Bifurcations of the FitzHugh–Nagumo Neuron Model

The dynamics of neurons consist of oscillating patterns of a membrane potential that underpin the operation of biological intelligence. The FitzHugh–Nagumo (FHN) model for neuron excitability generates rich dynamical regimes with a simpler mathematical structure than the Hodgkin–Huxley model. Because neurons can be understood in terms of electrical and electrochemical methods, here we apply the analysis of the impedance response to obtain the characteristic spectra and their evolution as a function of applied voltage. We convert the two nonlinear differential equations of FHN into an equivalent circuit model, classify the different impedance spectra, and calculate the corresponding trajectories in the phase plane of the variables. In analogy to the field of electrochemical oscillators, impedance spectroscopy detects the Hopf bifurcations and the spiking regimes. We show that a neuron element needs three essential internal components: capacitor, inductor, and negative differential resistance. The method supports the fabrication of memristor-based artificial neural networks.


Expression of FitzHugh-Nagumo (FHN) dynamical systems
There are different equivalent expressions of the FHN model. We present the main two representative parametrizations and our own formulation The expressions (1) and (2) are related by a linear transformation of the variables given in p. 12 of the book 3 .
Our expression (3) of the dynamical system has been selected so that the physical dimension of each variable is explicit. In this way the electrical quantities of the equivalent circuit are transparent and furthermore the results can be compared with experiments. Therefore, we have in (3) is time is a voltage is electrical current.
It follows that the constants have the dimensions , , are times 1 is voltage is electrical current. , are electrical resistances.
The use of and in the equation for the voltage, 3a, as well as the associated capacitance, = / , is quite extended, 4-6 since the original interpretation of the Hodgkin-Huxley model in terms of electrical circuits. 7 Therefore our equation for is quite conventional. In the equation 3b for we have used a similar structure for the definition of parameters , that generate the inductance = . The number of parameters in our model can be reduced to those that control de dynamical properties by defining dimensionless variables, as we have done in Eq. (7) and (8) of the main text: However, we prefer to use all the initial parameters so that the meaning of the circuit elements is very easily understood.
In the main text we have set 1 = 1 to simplify the calculations. This restriction is effect is a rescaling of the voltage scale. A general 1 needs to be restored when comparing with experiments.  ∆< 0 corresponds to real eigenvalues 1,2 with opposite sign. This is the region of negative , the three fixed points are a saddle and two sinks. The line = is a pitchfork bifurcation. On the other hand / > 1 corresponds to the single valued − , Fig. 1c. Here the stability of the fixed point is determined by < 0. At = 0 is the Hopf bifurcation. When > 0 and 2 − 4∆< 0 the fixed point becomes an unstable source with a pair of complex conjugate 1,2 . Summary: We analyze the dependence of the EC elements with respect to frequency, noting that small frequencies correspond to the dynamics of longer times, and viceversa. In Fig. S2 we calculate a total resistance and capacitance. For the simple arc spectrum in Fig. S2a the resistance increases suddenly at the characteristic frequency/time of the relaxation, given by or its inverse. In the spectrum with an inductive arc and positive resistance, 8 Fig. S2b, the low frequency resistance is smaller than the intermediate frequency resistance which has been used to explain hysteresis effects in solar cells. 9 In the oscillatory circuit S2c the variation of resistance is inverted with respect to b: a minimum of negative resistance that turns to positive at long times or low frequencies. Note that both (b) and (c) display a region of effective negative capacitance at low frequency. 8,10